Algebraic Proof for the Units and Tens Method

The Units and Tens Method is also known as the two finger method and uses .
First we will look at a three digit number multiplied by a single digit number.
We will refer to our three digit number as and the single digit multiplier as .

So multiplying this number by we get:

Expanding this out we get:

The , and are all pairs of single digit numbers. When multiplying two single digit numbers we normally get a two digit result, the exceptions are for the very low value digits where the result is a single digit number. However, these single digit results can be treated as a two digit result by adding a leading zero.
For example.

Two digit numbers can be represented like this:

Where is the tens digit, the digit multiplied by ten, and is the units digit, the digit multiplied by one.

We will use the letter to represent the tens digit and to represent the units digit of our two digit numbers.

As we have seen with our three digit number multiplied by a single digit number we get three pairs of numbers multiplied together so we need to include a subscript with our and so we can keep track of which pair they belong to.
We will use the following:

Where represents the tens digit of the result when is multiplied by and represents the unit digit of that result.

Putting this into our equation and expanding the parentheses we get:

This describes the method for the units and tens multiplication.
Looking at the term :

units of times the multiplier tens of times the multiplier

The normal way we write out our equation is :

We will place the and above the digits they refer to

Once in place the subscripts are not needed so we can simply put:

The * indicates which figure in the answer the and gives us.
The rest of the answer comes from the other terms in the equation in exactly the same way.
Thus we have proved the method of units and tens multiplication when multiplying by a single digit.